← Back to Avadari

The Role of $k(n)$ in the $\beta - \eta$ Bridge at the $s = 1$ Pole Boundary

1. The Coordinate Bridge

Let the Dirichlet Beta function and Dirichlet Eta function be defined on $\text{Re}(s) > 0$ by:

$$\beta(s) = \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^s}, \quad \eta(s) = \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k^s} = (1 - 2^{1-s})\zeta(s)$$

The Coordinate Ratio $K(s)$ and Coordinate Remainder $k(s)$ are defined by:

$$K(s) = 2^s \frac{\beta(s)}{\eta(s)}$$ $$k(s) = K(s) - 2^s = 2^s \left( \frac{\beta(s)}{\eta(s)} - 1 \right)$$

Inverting this coordinate mapping expresses Dirichlet Beta directly through $\eta(s)$ and the remainder $k(s)$:

$$\beta(s) = 2^{-s} \left( 2^s + k(s) \right) \eta(s) = 2^{-s} K(s) \, \eta(s)$$

2. Evaluation at the $s = 1$ Singularity

At $s = 1$:

1. The alternating harmonic series evaluates to: $$\eta(1) = \ln(2) \approx 0.6931471806$$
2. The Riemann zeta function has a simple pole with Laurent expansion $\zeta(s) = \frac{1}{s-1} + \gamma + \mathcal{O}(s-1)$, while the prefactor vanishes as: $$(1 - 2^{1-s}) = (s-1)\ln(2) - \frac{(s-1)^2}{2}\ln^2(2) + \mathcal{O}((s-1)^3)$$
3. The Coordinate Ratio $K(1)$ resolves the indeterminate form $0 \cdot \infty$ via L'Hôpital's rule: $$K(1) = \lim_{s \to 1} 2^s \frac{\beta(s)}{(1 - 2^{1-s})\zeta(s)} = 2^1 \frac{\pi/4}{\ln(2)} = \frac{\pi}{2\ln(2)} = \frac{\pi}{\ln(4)} \approx 2.2661800710$$
4. The Coordinate Remainder $k(1)$ is given by: $$k(1) = K(1) - 2^1 = \frac{\pi}{2\ln(2)} - 2 \approx 0.2661800710$$

3. Logarithmic Scale Cancellation

Substituting these evaluations into the bridge relation at $s = 1$:

$$\beta(1) = 2^{-1} \cdot K(1) \cdot \eta(1)$$ $$\beta(1) = \frac{1}{2} \left( \frac{\pi}{2\ln(2)} \right) \ln(2)$$

The $\ln(2)$ term in the numerator (from the alternating series sum $\eta(1)$) cancels with the $\ln(2)$ term in the denominator (generated by the derivative of the $(2^s - 2)$ prefactor at $s=1$):

$$\beta(1) = \frac{\pi}{4} \left( \frac{\ln(2)}{\ln(2)} \right) = \frac{\pi}{4}$$

Thus, $\pi/4$ represents the balanced state of the $k(s)$ coordinate bridge where logarithmic dilation from the alternating series is neutralized by the inverse scale of the pole prefactor.